how many positive integers m are there such that m^2 + 2017 is a perfect square?

[nanase]

the answer said 1 only, but I am interested in finding why.
I tried by using difference of two squares and moving stuff around like below but still feeling lost, please help thanks.

 

[BigBeachBanana]

View attachment 36675
the answer said 1 only, but I am interested in finding why.
I tried by using difference of two squares and moving stuff around like below but still feeling lost, please help thanks.
View attachment 36677

Hint: [imath]2017[/imath] is a prime number.

 

[nanase]

Hint: [imath]2017[/imath] is a prime number.

hi BBB, I check it is a prime number, but how does that help me in solving? does it mean there can only be one number that can be added to a prime number to make it square number?

 

[BigBeachBanana]

hi BBB, I check it is a prime number, but how does that help me in solving? does it mean there can only be one number that can be added to a prime number to make it square number?

Pick up from here [imath](x+m)(x-m)=2017[/imath] and use the fact that [imath]2017[/imath] is prime.

 

[nanase]

I am sorry BBB I am very weak in Maths, I am not getting how that helps me, sorry sorry T-T can you re-explain

 

[BigBeachBanana]

I am sorry BBB I am very weak in Maths, I am not getting how that helps me, sorry sorry T-T can you re-explain

What 2 numbers multiplied together to make 2017 if 2017 is prime?

 

[blamocur]

I am sorry BBB I am very weak in Maths, I am not getting how that helps me, sorry sorry T-T can you re-explain

If you represent a prime number as a product of two numbers what can you tell about those numbers?

 

[nanase]

holy moly BBB and blamocur, on some paper scribbling now I got it
x+m must be 2017
x-m must be 1
solving simultaneously m equals to 1008
thankssss so much guys

 

[BigBeachBanana]

holy moly BBB and blamocur, on some paper scribbling now I got it
x+m must be 2017
x-m must be 1
solving simultaneously m equals to 1008
thankssss so much guys

For completeness, you should also consider x+m = 1 and x-m=2017 since multiplication is commutative.

 

[averagestudentinph]

holy moly BBB and blamocur, on some paper scribbling now I got it
x+m must be 2017
x-m must be 1
solving simultaneously m equals to 1008
thankssss so much guys

Hii!! Can I ask how did y'all solve for 1008? Im a bit lost

 

[khansaheb]

Hii!! Can I ask how did y'all solve for 1008? Im a bit lost

x + m = 2017 .................................. (1)
x - m = 1 ...........................................(2)

add (1) & (2)

2 * x = 2018 → x= 2018/2 =1009.............................and

subtract (2) from (1)

2 * m = 2016 → m= ?.............................and

I believe you can now answer your own question now!!